Part IB. Quantum Mechanics. Year


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1 Part IB Year
2 Paper 4, Section I 6B (a) Define the quantum orbital angular momentum operator ˆL = (ˆL 1, ˆL 2, ˆL 3 ) in three dimensions, in terms of the position and momentum operators. (b) Show that [ˆL 1, ˆL 2 ] = i ˆL 3. [You may assume that the position and momentum operators satisfy the canonical commutation relations.] (c) Let ˆL 2 = ˆL ˆL ˆL 2 3. Show that ˆL 1 commutes with ˆL 2. [In this part of the question you may additionally assume without proof the permuted relations [ˆL 2, ˆL 3 ] = i ˆL 1 and [ˆL 3, ˆL 1 ] = i ˆL 2.] [Hint: It may be useful to consider the expression [Â, ˆB] ˆB + ˆB [Â, ˆB] for suitable operators Â and ˆB.] (d) Suppose that ψ 1 (x, y, z) and ψ 2 (x, y, z) are normalised eigenstates of ˆL 1 with eigenvalues and respectively. Consider the wavefunction ψ = ψ 1 cos ωt + 2 ψ 2 sin ωt, with ω being a positive constant. Find the earliest time t 0 > 0 such that the expectation value of ˆL 1 in ψ is zero. Paper 3, Section I 8B (a) Consider a quantum particle moving in one space dimension, in a timeindependent real potential V (x). For a wavefunction ψ(x, t), define the probability density ρ(x, t) and probability current j(x, t) and show that ρ t + j x = 0. (b) Suppose now that V (x) = 0 and ψ(x, t) = (e ikx + Re ikx )e iet/, where E = 2 k 2 /(2m), k and m are real positive constants, and R is a complex constant. Compute the probability current for this wavefunction. Interpret the terms in ψ and comment on how this relates to the computed expression for the probability current. Part IB, 2016 List of Questions [TURN OVER
3 Paper 1, Section II 15B (a) A particle of mass m in one space dimension is confined to move in a potential V (x) given by { 0 for 0 < x < a, V (x) = for x < 0 or x > a. The normalised initial wavefunction of the particle at time t = 0 is ψ 0 (x) = 4 ( sin 3 πx ). 5a a (i) Find the expectation value of the energy at time t = 0. (ii) Find the wavefunction of the particle at time t = 1. [Hint: It may be useful to recall the identity sin 3θ = 3 sin θ 4 sin 3 θ.] (b) The right hand wall of the potential is lowered to a finite constant value U 0 > 0 giving the new potential: 0 for 0 < x < a, U(x) = for x < 0, U 0 for x > a. This potential is set up in the laboratory but the value of U 0 is unknown. The stationary states of the potential are investigated and it is found that there exists exactly one bound state. Show that the value of U 0 must satisfy π 2 2 8ma 2 < U 0 < 9π2 2 8ma 2. Part IB, 2016 List of Questions
4 Paper 3, Section II 16B The spherically symmetric bound state wavefunctions ψ(r) for the Coulomb potential V = e 2 /(4πǫ 0 r) are normalisable solutions of the equation d 2 ψ dr dψ r dr + 2λ r ψ = 2mE 2 ψ. Here λ = (me 2 )/(4πǫ 0 2 ) and E < 0 is the energy of the state. (a) By writing the wavefunction as ψ(r) = f(r) exp( Kr), for a suitable constant K that you should determine, show that there are normalisable wavefunctions ψ(r) only for energies of the form me 4 E = 32π 2 ǫ N 2, with N being a positive integer. (b) The energies in (a) reproduce the predictions of the Bohr model of the hydrogen atom. How do the wavefunctions above compare to the assumptions in the Bohr model? Part IB, 2016 List of Questions [TURN OVER
5 2016 Paper 2, Section II 17B The one dimensional quantum harmonic oscillator has Hamiltonian 44 Ĥ = 1 2m ˆp mω2ˆx 2, where m and ω are real positive constants and ˆx and ˆp are the standard position and momentum operators satisfying the commutation relation [ˆx, ˆp] = i. Consider the operators Â = ˆp imωˆx and ˆB = ˆp + imωˆx. (a) Show that ˆBÂ = 2m ( Ĥ 1 2 ω ) and Â ˆB = 2m (Ĥ + 12 ) ω. (b) Suppose that φ is an eigenfunction of Ĥ with eigenvalue E. Show that then also an eigenfunction of Ĥ and that its corresponding eigenvalue is E ω. (c) Show that for any normalisable wavefunctions χ and ψ, Âφ is χ (Âψ) dx = ( ˆBχ) ψ dx. [You may assume that the operators ˆx and ˆp are Hermitian.] (d) With φ as in (b), obtain an expression for the norm of Âφ in terms of E and the norm of φ. [The squared norm of any wavefunction ψ is ψ 2 dx.] (e) Show that all eigenvalues of Ĥ are nonnegative. (f) Using the above results, deduce that each eigenvalue E of Ĥ must be of the form E = (n ) ω for some nonnegative integer n. Part IB, 2016 List of Questions
6 Paper 4, Section I 6D The radial wavefunction R(r) for an electron in a hydrogen atom satisfies the equation 2 d 2mr 2 dr (r 2 ddr ) R(r) + 2 l(l + 1)R(r) e2 2mr2 R(r) = E R(r) 4πǫ 0 r ( ) Briefly explain the origin of each term in this equation. The wavefunctions for the ground state and the first radially excited state, both with l = 0, can be written as R 1 (r) = N 1 e αr R 2 (r) = N 2 ( rα) e 1 2 αr where N 1 and N 2 are normalisation constants. Verify that R 1 (r) is a solution of ( ), determining α and finding the corresponding energy eigenvalue E 1. Assuming that R 2 (r) is a solution of ( ), compare coefficients of the dominant terms when r is large to determine the corresponding energy eigenvalue E 2. [You do not need to find N 1 or N 2, nor show that R 2 is a solution of ( ).] A hydrogen atom makes a transition from the first radially excited state to the ground state, emitting a photon. What is the angular frequency of the emitted photon? Paper 3, Section I 8D A quantummechanical system has normalised energy eigenstates χ 1 and χ 2 with nondegenerate energies E 1 and E 2 respectively. The observable A has normalised eigenstates, φ 1 = C(χ 1 + 2χ 2 ), eigenvalue = a 1, φ 2 = C(2χ 1 χ 2 ), eigenvalue = a 2, where C is a positive real constant. Determine C. Initially, at time t = 0, the state of the system is φ 1. Write down an expression for ψ(t), the state of the system with t 0. What is the probability that a measurement of energy at time t will yield E 2? For the same initial state, determine the probability that a measurement of A at time t > 0 will yield a 1 and the probability that it will yield a 2. Part IB, 2015 List of Questions
7 Paper 1, Section II 15D Write down expressions for the probability density ρ(x, t) and the probability current j(x, t) for a particle in one dimension with wavefunction Ψ(x, t). If Ψ(x, t) obeys the timedependent Schrödinger equation with a real potential, show that j x + ρ t = 0. Consider a stationary state, Ψ(x, t) = ψ(x)e iet/, with { e ik 1 ψ(x) x + Re ik 1x x T e ik 2x x +, where E, k 1, k 2 are real. x. Consider a real potential, Evaluate j(x, t) for this state in the regimes x + and V (x) = αδ(x) + U(x), U(x) = { 0 x < 0 V 0 x > 0, where δ(x) is the Dirac delta function, V 0 > 0 and α > 0. Assuming that ψ(x) is continuous at x = 0, derive an expression for [ lim ψ (ǫ) ψ ( ǫ) ]. ǫ 0 Hence calculate the reflection and transmission probabilities for a particle incident from x = with energy E > V 0. Part IB, 2015 List of Questions [TURN OVER
8 Paper 3, Section II 16D Define the angular momentum operators ˆL i for a particle in three dimensions in terms of the position and momentum operators ˆx i and ˆp i = i Write down an expression for [ˆL i, ˆL j ] and use this to show that [ˆL 2, ˆL i ] = 0 where ˆL 2 = ˆL 2 x + ˆL 2 y + ˆL 2 z. What is the significance of these two commutation relations? Let ψ(x, y, z) be both an eigenstate of ˆL z with eigenvalue zero and an eigenstate of ˆL 2 with eigenvalue 2 l(l + 1). Show that (ˆL x + iˆl y )ψ is also an eigenstate of both ˆL z and ˆL 2 and determine the corresponding eigenvalues. Find real constants A and B such that x i. φ(x, y, z) = ( Az 2 + By 2 r 2) e r, r 2 = x 2 + y 2 + z 2, is an eigenfunction of ˆL z with eigenvalue zero and an eigenfunction of ˆL 2 with an eigenvalue which you should determine. [Hint: You might like to show that ˆL i f(r) = 0.] Paper 2, Section II 17D A quantummechanical harmonic oscillator has Hamiltonian where k is a positive real constant. operators. The eigenfunctions of ( ) can be written as Ĥ = ˆp k2ˆx 2. ( ), ψ n (x) = h n (x ) k/ Show that ˆx = x and ˆp = i x exp ) ( kx2, 2 are Hermitian where h n is a polynomial of degree n with even (odd) parity for even (odd) n and n = 0, 1, 2,.... Show that ˆx = ˆp = 0 for all of the states ψ n. State the Heisenberg uncertainty principle and verify it for the state ψ 0 by computing ( x) and ( p). [Hint: You should properly normalise the state.] The oscillator is in its ground state ψ 0 when the potential is suddenly changed so that k 4k. If the wavefunction is expanded in terms of the energy eigenfunctions of the new Hamiltonian, φ n, what can be said about the coefficient of φ n for odd n? What is the probability that the particle is in the new ground state just after the change? [Hint: You may assume that if I n = e ax2 x n dx then I 0 = π a and I 2 = 1 2a π a.] Part IB, 2015 List of Questions
9 Paper 4, Section I 6A For some quantum mechanical observable Q, prove that its uncertainty ( Q) satisfies ( Q) 2 = Q 2 Q 2. A quantum mechanical harmonic oscillator has Hamiltonian H = p2 2m + mω2 x 2, 2 where m > 0. Show that (in a stationary state of energy E) E ( p)2 2m + mω2 ( x) 2. 2 Write down the Heisenberg uncertainty relation. Then, use it to show that for our stationary state. E 1 2 ω Paper 3, Section I 8A The wavefunction of a normalised Gaussian wavepacket for a particle of mass m in one dimension with potential V (x) = 0 is given by ψ(x, t) = B ( x 2 ) A(t) A(t) exp, 2 where A(0) = 1. Given that ψ(x, t) is a solution of the timedependent Schrödinger equation, find the complexvalued function A(t) and the real constant B. [You may assume that e λx2 dx = π/ λ.] Part IB, 2014 List of Questions
10 Paper 1, Section II 15A Consider a particle confined in a onedimensional infinite potential well: V (x) = for x a and V (x) = 0 for x < a. The normalised stationary states are ( ) πn(x + a) α n sin for x < a ψ n (x) = 2a 0 for x a where n = 1, 2,.... (i) Determine the α n and the stationary states energies E n. (ii) A state is prepared within this potential well: ψ(x) x for 0 < x < a, but ψ(x) = 0 for x 0 or x a. Find an explicit expansion of ψ(x) in terms of ψ n (x). (iii) If the energy of the state is then immediately measured, show that the probability that it is greater than 2 π 2 is ma 2 4 n=0 b n π n, where the b n are integers which you should find. (iv) By considering the normalisation condition for ψ(x) in terms of the expansion in ψ n (x), show that π 2 3 = p=1 A p 2 + ) B 2 (1 (2p 1) 2 + C( 1)p, (2p 1)π where A, B and C are integers which you should find. Part IB, 2014 List of Questions [TURN OVER
11 2014 Paper 3, Section II 16A The Hamiltonian of a twodimensional isotropic harmonic oscillator is given by H = p2 x + p 2 y 2m 40 + mω2 2 (x2 + y 2 ), where x and y denote position operators and p x and p y the corresponding momentum operators. State without proof the commutation relations between the operators x, y, p x, p y. From these commutation relations, write [x 2, p x ] and [x, p 2 x ] in terms of a single operator. Now consider the observable L = xp y yp x. Ehrenfest s theorem states that, for some observable Q with expectation value Q, d Q dt = 1 [Q, H] + Q i t. Use it to show that the expectation value of L is constant with time. Given two states ψ 1 = αx exp ( β(x 2 + y 2 ) ) and ψ 2 = αy exp ( β(x 2 + y 2 ) ), where α and β are constants, find a normalised linear combination of ψ 1 and ψ 2 that is an eigenstate of L, and the corresponding L eigenvalue. [You may assume that α correctly normalises both ψ 1 and ψ 2.] If a quantum state is prepared in the linear combination you have found at time t = 0, what is the expectation value of L at a later time t? Part IB, 2014 List of Questions
12 Paper 2, Section II 17A For an electron of mass m in a hydrogen atom, the timeindependent Schrödinger equation may be written as 2 2mr 2 r ( r 2 ψ ) + 1 r 2mr 2 L2 ψ Consider normalised energy eigenstates of the form ψ lm (r, θ, φ) = R(r)Y lm (θ, φ) where Y lm are orbital angular momentum eigenstates: π θ=0 e2 4πǫ 0 r ψ = Eψ. L 2 Y lm = 2 l(l + 1)Y lm, L 3 Y lm = my lm, where l = 1, 2,... and m = 0, ±1, ±2,... ± l. The Y lm functions are normalised with 2π φ=0 Y lm 2 sin θ dθ dφ = 1. (i) Write down the resulting equation satisfied by R(r), for fixed l. Show that it has solutions of the form ( R(r) = Ar l exp r ), a(l + 1) where a is a constant which you should determine. Show that e2 E = Dπǫ 0 a, where D is an integer which you should find (in terms of l). Also, show that A 2 = 2 2l+3 a F G!(l + 1) H, where F, G and H are integers that you should find in terms of l. (ii) Given the radius of the proton r p a, show that the probability of the electron being found within the proton is approximately 2 2l+3 finding the integer C in terms of l. [You may assume that 0 t l e t dt = l!.] C ( rp ) 2l+3 [ ( rp )] 1 + O, a a Part IB, 2014 List of Questions [TURN OVER
13 Paper 4, Section I 6B The components of the threedimensional angular momentum operator ˆL are defined as follows: ( ˆL x = i y z z ) ( ˆLy = i z y x x ) ˆLz = i ( x z y y ). x Given that the wavefunction ψ = (f(x) + iy)z is an eigenfunction of ˆL z, find all possible values of f(x) and the corresponding eigenvalues of ψ. Letting f(x) = x, show that ψ is an eigenfunction of ˆL 2 and calculate the corresponding eigenvalue. Paper 3, Section I 8B If α, β and γ are linear operators, establish the identity [αβ, γ] = α[β, γ] + [α, γ]β. In what follows, the operators x and p are Hermitian and represent position and momentum of a quantum mechanical particle in onedimension. Show that and [x n, p] = i nx n 1 [x, p m ] = i mp m 1 where m, n Z +. Assuming [x n, p m ] 0, show that the operators x n and p m are Hermitian but their product is not. Determine whether x n p m + p m x n is Hermitian. Part IB, 2013 List of Questions
14 Paper 1, Section II 15B A particle with momentum ˆp moves in a onedimensional real potential with Hamiltonian given by Ĥ = 1 (ˆp + isa)(ˆp isa), 2m < x < where A is a real function and s R +. Obtain the potential energy of the system. Find χ(x) such that (ˆp isa)χ(x) = 0. Now, putting A = x n, for n Z +, show that χ(x) can be normalised only if n is odd. Letting n = 1, use the inequality to show that assuming that both ˆp and ˆx vanish. ψ (x)ĥψ(x)dx 0 x p 2 Paper 3, Section II 16B Obtain, with the aid of the timedependent Schrödinger equation, the conservation equation ρ(x, t) + j(x, t) = 0 t where ρ(x, t) is the probability density and j(x, t) is the probability current. What have you assumed about the potential energy of the system? Show that if the potential U(x, t) is complex the conservation equation becomes t ρ(x, t) + j(x, t) = 2 ρ(x, t) ImU(x, t). Take the potential to be timeindependent. Show, with the aid of the divergence theorem, that d ρ(x, t) dv = 2 ρ(x, t) ImU(x) dv. dt R 3 R 3 Assuming the wavefunction ψ(x, 0) is normalised to unity, show that if ρ(x, t) is expanded about t = 0 so that ρ(x, t) = ρ 0 (x) + tρ 1 (x) +, then ρ(x, t) dv = 1 + 2t ρ 0 (x) ImU(x) dv +. R 3 R 3 As time increases, how does the quantity on the left of this equation behave if ImU(x) < 0? Part IB, 2013 List of Questions [TURN OVER
15 Paper 2, Section II 17B (i) Consider a particle of mass m confined to a onedimensional potential well of depth U > 0 and potential { U, x < l V (x) = 0, x > l. If the particle has energy E where U E < 0, show that for even states α tan αl = β where α = [ 2m 2 (U + E)] 1/2 and β = [ 2m 2 E] 1/2. (ii) A particle of mass m that is incident from the left scatters off a onedimensional potential given by V (x) = kδ(x) where δ(x) is the Dirac delta. If the particle has energy E > 0 and k > 0, obtain the reflection and transmission coefficients R and T, respectively. Confirm that R + T = 1. For the case k < 0 and E < 0 show that the energy of the only even parity bound state of the system is given by E = mk Use part (i) to verify this result by taking the limit U, l 0 with Ul fixed. Part IB, 2013 List of Questions
16 2012 Paper 4, Section I 6C In terms of quantum states, what is meant by energy degeneracy? 39 A particle of mass m is confined within the box 0 < x < a, 0 < y < a and 0 < z < c. The potential vanishes inside the box and is infinite outside. Find the allowed energies by considering a stationary state wavefunction of the form χ(x, y, z) = X(x) Y (y) Z(z). Write down the normalised ground state wavefunction. Assuming that c < a < 2c, give the energies of the first three excited states. Paper 3, Section I 8C A onedimensional quantum mechanical particle has normalised bound state energy eigenfunctions χ n (x) and corresponding nondegenerate energy eigenvalues E n. At t = 0 the normalised wavefunction ψ(x, t) is given by ψ(x, 0) = 5 6 eik 1 1 χ 1 (x) + 6 eik 2 χ 2 (x) where k 1 and k 2 are real constants. Write down the expression for ψ(x, t) at a later time t and give the probability that a measurement of the particle s energy will yield a value of E 2. Show that the expectation value of x at time t is given by x = 5 6 x [ ] 6 x Re x 12 e i(k 2 k 1 ) i(e 2 E 1 )t/ where x ij = χ i (x) x χ j(x) dx. Part IB, 2012 List of Questions [TURN OVER
17 Paper 1, Section II 15C Show that if the energy levels are discrete, the general solution of the Schrödinger equation i ψ t = 2 2m 2 ψ + V (x)ψ is a linear superposition of stationary states ψ(x, t) = a n χ n (x) exp( ie n t/ ), n=1 where χ n (x) is a solution of the timeindependent Schrödinger equation and a n are complex coefficients. Can this general solution be considered to be a stationary state? Justify your answer. A linear operator Ô acts on the orthonormal energy eigenfunctions χ n as follows: Ôχ 1 = χ 1 + χ 2 Ôχ 2 = χ 1 + χ 2 Ôχ n = 0, n 3. Obtain the eigenvalues of Ô. Hence, find the normalised eigenfunctions of Ô. In an experiment a measurement is made of Ô at t = 0 yielding an eigenvalue of 2. What is the probability that a measurement at some later time t will yield an eigenvalue of 2? Paper 3, Section II 16C State the condition for a linear operator Ô to be Hermitian. Given the position and momentum operators ˆx i and ˆp i = i x i, define the angular momentum operators ˆL i. Establish the commutation relations [ˆL i, ˆL j ] = i ǫ ijk ˆLk and use these relations to show that ˆL 3 is Hermitian assuming ˆL 1 and ˆL 2 are. Consider a wavefunction of the form χ(x) = x 3 (x 1 + kx 2 )e r where r = x and k is some constant. Show that χ(x) is an eigenstate of the total angular momentum operator ˆL 2 for all k, and calculate the corresponding eigenvalue. For what values of k is χ(x) an eigenstate of ˆL 3? What are the corresponding eigenvalues? Part IB, 2012 List of Questions
18 2012 Paper 2, Section II 17C 41 Consider a quantum mechanical particle in a onedimensional potential V (x), for which V (x) = V ( x). Prove that when the energy eigenvalue E is nondegenerate, the energy eigenfunction χ(x) has definite parity. Now assume the particle is in the double potential well U, 0 x l 1 V (x) = 0, l 1 < x l 2, l 2 < x, where 0 < l 1 < l 2 and 0 < E < U (U being large and positive). Obtain general expressions for the even parity energy eigenfunctions χ + (x) in terms of trigonometric and hyperbolic functions. Show that where k 2 = 2mE 2 and κ 2 2m(U E) = 2. tan[k(l 2 l 1 )] = k κ coth(κl 1), Part IB, 2012 List of Questions [TURN OVER
19 Paper 3, Section I 8C A particle of mass m and energy E, incident from x =, scatters off a delta function potential at x = 0. The time independent Schrödinger equation is 2 d 2 ψ + Uδ(x)ψ = Eψ 2m dx2 where U is a positive constant. Find the reflection and transmission probabilities. Paper 4, Section I 6C Consider the 3dimensional oscillator with Hamiltonian H = 2 2m 2 + mω2 2 (x2 + y 2 + 4z 2 ). Find the ground state energy and the spacing between energy levels. Find the degeneracies of the lowest three energy levels. [You may assume that the energy levels of the 1dimensional harmonic oscillator with Hamiltonian H 0 = 2 d 2 2m dx 2 + mω2 2 x2 are (n ) ω, n = 0, 1, 2,....] Part IB, 2011 List of Questions [TURN OVER
20 Paper 1, Section II 15C For a quantum mechanical particle moving freely on a circle of length 2π, the wavefunction ψ(t, x) satisfies the Schrödinger equation i ψ t = 2 2 ψ 2m x 2 on the interval 0 x 2π, and also the periodicity conditions ψ(t, 2π) = ψ(t, 0), and ψ ψ (t, 2π) = (t, 0). Find the allowed energy levels of the particle, and their degeneracies. x x The current is defined as j = i ( ψ ψ ψ ψ ) 2m x x where ψ is a normalized state. Write down the general normalized state of the particle when it has energy 2 2 /m, and show that in any such state the current j is independent of x and t. Find a state with this energy for which the current has its maximum positive value, and find a state with this energy for which the current vanishes. Paper 2, Section II 17C The quantum mechanical angular momentum operators are L i = i ǫ ijk x j Show that each of these is hermitian. x k (i = 1, 2, 3). The total angular momentum operator is defined as L 2 = L L2 2 + L2 3. Show that L 2 L 2 3 in any state, and show that the only states where L2 = L 2 3 are those with no angular dependence. Verify that the eigenvalues of the operators L 2 and L 2 3 (whose values you may quote without proof) are consistent with these results. Part IB, 2011 List of Questions
21 Paper 3, Section II 16C For an electron in a hydrogen atom, the stationary state wavefunctions are of the form ψ(r, θ, φ) = R(r)Y lm (θ, φ), where in suitable units R obeys the radial equation d 2 R dr ( dr l(l + 1) r dr r 2 R + 2 E + 1 ) R = 0. r Explain briefly how the terms in this equation arise. This radial equation has bound state solutions of energy E = E n, where E n = 1 (n = 1, 2, 3,... ). Show that when l = n 1, there is a solution of the 2n 2 form R(r) = r α e r/n, and determine α. Find the expectation value r in this state. What is the total degeneracy of the energy level with energy E n? Part IB, 2011 List of Questions [TURN OVER
22 Paper 3, Section I 8D Write down the commutation relations between the components of position x and momentum p for a particle in three dimensions. A particle of mass m executes simple harmonic motion with Hamiltonian H = 1 2m p2 + mω2 2 x2, and the orbital angular momentum operator is defined by L = x p. Show that the components of L are observables commuting with H. Explain briefly why the components of L are not simultaneous observables. What are the implications for the labelling of states of the threedimensional harmonic oscillator? Paper 4, Section I 6D Determine the possible values of the energy of a particle free to move inside a cube of side a, confined there by a potential which is infinite outside and zero inside. What is the degeneracy of the lowestbutone energy level? Part IB, 2010 List of Questions [TURN OVER
23 2010 Paper 1, Section II 15D A particle of unit mass moves in one dimension in a potential 38 V = 1 2 ω2 x 2. Show that the stationary solutions can be written in the form ψ n (x) = f n (x) exp( αx 2 ). You should give the value of α and derive any restrictions on f n (x). Hence determine the possible energy eigenvalues E n. The particle has a wave function ψ(x, t) which is even in x at t = 0. Write down the general form for ψ(x, 0), using the fact that f n (x) is an even function of x only if n is even. Hence write down ψ(x, t) and show that its probability density is periodic in time with period π/ω. Paper 2, Section II 17D A particle of mass m moves in a onedimensional potential defined by for x < 0, V (x) = 0 for 0 x a, V 0 for a < x, where a and V 0 are positive constants. Defining c = [2m(V 0 E)] 1/2 / and k = (2mE) 1/2 /, show that for any allowed positive value E of the energy with E < V 0 then c + k cot ka = 0. Find the minimum value of V 0 for this equation to have a solution. Find the normalized wave function for the particle. Write down an expression for the expectation value of x in terms of two integrals, which you need not evaluate. Given that x = 1 (ka tan ka), 2k discuss briefly the possibility of x being greater than a. [Hint: consider the graph of ka cot ka against ka.] Part IB, 2010 List of Questions
24 Paper 3, Section II 16D A π (a particle of the same charge as the electron but 270 times more massive) is bound in the Coulomb potential of a proton. Assuming that the wave function has the form ce r/a, where c and a are constants, determine the normalized wave function of the lowest energy state of the π, assuming it to be an Swave (i.e. the state with l = 0). (You should treat the proton as fixed in space.) Calculate the probability of finding the π inside a sphere of radius R in terms of the ratio µ = R/a, and show that this probability is given by 4µ 3 /3 + O(µ 4 ) if µ is very small. Would the result be larger or smaller if the π were in a P wave (l = 1) state? Justify your answer very briefly. [Hint: in spherical polar coordinates, 2 ψ(r) = 1 r 2 r 2 (rψ) + 1 r 2 sin θ ( sin θ ψ ) + θ θ 1 2 ] ψ r 2 sin 2 θ φ 2. Part IB, 2010 List of Questions [TURN OVER
25 2009 Paper 3, Section I 7B 42 The motion of a particle in one dimension is described by the timeindependent hermitian Hamiltonian operator H whose normalized eigenstates ψ n (x), n = 0, 1, 2,..., satisfy the Schrödinger equation Hψ n = E n ψ n, with E 0 < E 1 < E 2 < < E n <. Show that ψ mψ n dx = δ mn. The particle is in a state represented by the wavefunction Ψ(x, t) which, at time t = 0, is given by ( ) 1 n+1 Ψ(x, 0) = ψ n (x). 2 Write down an expression for Ψ(x, t) and show that it is normalized to unity. n=0 Derive an expression for the expectation value of the energy for this state and show that it is independent of time. Calculate the probability that the particle has energy E m for a given integer m 0, and show that this also is timeindependent. Paper 4, Section I 6B The wavefunction of a Gaussian wavepacket for a particle of mass m moving in one dimension is ψ(x, t) = 1 ( 1 π 1/4 1 + i t/m exp x 2 ). 2(1 + i t/m) Show that ψ(x, t) satisfies the appropriate timedependent Schrödinger equation. Show that ψ(x, t) is normalized to unity and calculate the uncertainty in measurement of the particle position, x = x 2 x 2. Is ψ(x, t) a stationary state? Give a reason for your answer. [ You may assume that e λx2 dx = ] π λ. Part IB, 2009 List of Questions
26 2009 Paper 1, Section II 15B 43 A particle of mass m moves in one dimension in a potential V (x) which satisfies V (x) = V ( x). Show that the eigenstates of the Hamiltonian H can be chosen so that they are also eigenstates of the parity operator P. For eigenstates with odd parity ψ odd (x), show that ψ odd (0) = 0. A potential V (x) is given by V (x) = { κδ(x) x < a x > a. State the boundary conditions satisfied by ψ(x) at x = a, and show also that 2 [ dψ 2m lim ǫ 0 dx dψ ] ǫ dx = κψ(0). ǫ Let the energy eigenstates of even parity be given by A cos λx + B sin λx a < x < 0 ψ even (x) = A cos λx B sin λx 0 < x < a 0 otherwise. Verify that ψ even (x) satisfies P ψ even (x) = ψ even (x). satisfy By demanding that ψ even (x) satisfy the relevant boundary conditions show that tan λa = 2 λ m κ. For κ > 0 show that the energy eigenvalues E even n Show also that η n = E even n 1 2m [ ] (2n + 1) π 2 > 0. 2a lim η n = 0, n and give a physical explanation of this result., n = 0, 1, 2,..., with E even n < E even n+1, Show that the energy eigenstates with odd parity and their energy eigenvalues do not depend on κ. Part IB, 2009 List of Questions [TURN OVER
27 2009 Paper 2, Section II 16B 44 Write down the expressions for the probability density ρ and the associated current density j for a particle with wavefunction ψ(x, t) moving in one dimension. If ψ(x, t) obeys the timedependent Schrödinger equation show that ρ and j satisfy j x + ρ t = 0. Give an interpretation of ψ(x, t) in the case that ψ(x, t) = (e ikx + Re ikx )e iet/, and show that E = 2 k 2 ρ and 2m t = 0. A particle of mass m and energy E > 0 moving in one dimension is incident from the left on a potential V (x) given by { V0 0 < x < a V (x) = 0 x < 0, x > a, where V 0 is a positive constant. What conditions must be imposed on the wavefunction at x = 0 and x = a? Show that when 3E = V 0 the probability of transmission is [ a ] 1 8mE 16 sin2. For what values of a does this agree with the classical result? Part IB, 2009 List of Questions
28 Paper 3, Section II 16B If A, B, and C are operators establish the identity [AB, C] = A[B, C] + [A, C]B. A particle moves in a twodimensional harmonic oscillator potential with Hamiltonian H = 1 2 (p2 x + p 2 y) (x2 + y 2 ). The angular momentum operator is defined by L = xp y yp x. Show that L is hermitian and hence that its eigenvalues are real. Establish the commutation relation [L, H] = 0. Why does this ensure that eigenstates of H can also be chosen to be eigenstates of L? Let φ 0 (x, y) = e (x2 +y 2 )/2, and show that φ 0, φ x = xφ 0 and φ y = yφ 0 are all eigenstates of H, and find their respective eigenvalues. Show that Lφ 0 = 0, Lφ x = i φ y, Lφ y = i φ x, and hence, by taking suitable linear combinations of φ x and φ y, find two states, ψ 1 and ψ 2, satisfying Lψ j = λ j ψ j, Hψ j = E j ψ j j = 1, 2. Show that ψ 1 and ψ 2 are orthogonal, and find λ 1, λ 2, E 1 and E 2. The particle has charge e, and an electric field of strength E is applied in the x direction so that the Hamiltonian is now H, where H = H eex. Show that [L, H ] = i eey. Why does this mean that L and H cannot have simultaneous eigenstates? By making the change of coordinates x = x ee, y = y, show that ψ 1 (x, y ) and ψ 2 (x, y ) are eigenstates of H and write down the corresponding energy eigenvalues. Find a modified angular momentum operator L for which ψ 1 (x, y ) and ψ 2 (x, y ) are also eigenstates. Part IB, 2009 List of Questions [TURN OVER
29 /II/15A The radial wavefunction g(r) for the hydrogen atom satisfies the equation 2 2mr 2 ( d r 2 dg(r) ) e2 g(r) dr dr 4πɛ 0 r l(l + 1) + 2 g(r) = Eg(r). ( ) 2mr2 With reference to the general form for the timeindependent Schrödinger equation, explain the origin of each term. What are the allowed values of l? The lowestenergy boundstate solution of ( ), for given l, has the form r α e βr. Find α and β and the corresponding energy E in terms of l. A hydrogen atom makes a transition between two such states corresponding to l+1 and l. What is the frequency of the emitted photon? 2/II/16A Give the physical interpretation of the expression A ψ = ψ(x) Âψ(x)dx for an observable A, where Â is a Hermitian operator and ψ is normalised. By considering the norm of the state (A + iλb)ψ for two observables A and B, and real values of λ, show that A 2 ψ B 2 ψ 1 4 [A, B] ψ 2. Deduce the uncertainty relation where A is the uncertainty of A. A B 1 2 [A, B] ψ, A particle of mass m moves in one dimension under the influence of potential 1 2 mω2 x 2. By considering the commutator [x, p], show that the expectation value of the Hamiltonian satisfies H ψ 1 2 ω. Part IB 2008
30 /I/7A Write down a formula for the orbital angular momentum operator ˆL. Show that its components satisfy [L i, L j ] = i ɛ ijk L k. If L 3 ψ = 0, show that (L 1 ± il 2 )ψ are also eigenvectors of L 3, and find their eigenvalues. 3/II/16A What is the probability current for a particle of mass m, wavefunction ψ, moving in one dimension? A particle of energy E is incident from x < 0 on a barrier given by 0 x 0 V (x) = V 1 0 < x < a x a V 0 where V 1 > V 0 > 0. What are the conditions satisfied by ψ at x = 0 and x = a? Write down the form taken by the wavefunction in the regions x 0 and x a distinguishing between the cases E > V 0 and E < V 0. For both cases, use your expressions for ψ to calculate the probability currents in these two regions. Define the reflection and transmission coefficients, R and T. Using current conservation, show that the expressions you have derived satisfy R + T = 1. Show that T = 0 if 0 < E < V 0. 4/I/6A What is meant by a stationary state? What form does the wavefunction take in such a state? A particle has wavefunction ψ(x, t), such that ψ(x, 0) = 1 2 (χ 1(x) + χ 2 (x)), where χ 1 and χ 2 are normalised eigenstates of the Hamiltonian with energies E 1 and E 2. Write down ψ(x, t) at time t. Show that the expectation value of A at time t is A ψ = 1 2 ( ) ) χ 1Âχ 1 + χ 2Âχ 2 dx + Re (e i(e1 E2)t/ χ 1Âχ 2 dx. Part IB 2008
31 /II/15B The relative motion of a neutron and proton is described by the Schrödinger equation for a single particle of mass m under the influence of the central potential V (r) = { U r < a 0 r > a, where U and a are positive constants. Solve this equation for a spherically symmetric state of the deuteron, which is a bound state of a proton and neutron, giving the condition on U for this state to exist. [If ψ is spherically symmetric then 2 ψ = 1 r d 2 dr 2 (rψ).] 2/II/16B Write down the angular momentum operators L 1, L 2, L 3 in terms of the position and momentum operators, x and p, and the commutation relations satisfied by x and p. Verify the commutation relations [L i, L j ] = i ɛ ijk L k. Further, show that [L i, p j ] = i ɛ ijk p k. A wavefunction Ψ 0 (r) is spherically symmetric. Verify that LΨ 0 (r) = 0. Consider the vector function Φ = Ψ 0 (r). Show that Φ 3 and Φ 1 ± iφ 2 are eigenfunctions of L 3 with eigenvalues 0, ± respectively. Part IB 2007
32 /I/7B The quantum mechanical harmonic oscillator has Hamiltonian H = 1 2m p mω2 x 2, and is in a stationary state of energy < H >= E. Show that E 1 2m ( p) mω2 ( x) 2, where ( p) 2 = p 2 p 2 and ( x) 2 = x 2 x 2. Use the Heisenberg Uncertainty Principle to show that E 1 2 ω. 3/II/16B A quantum system has a complete set of orthonormal eigenstates, ψ n (x), with nondegenerate energy eigenvalues, E n, where n = 1, 2, Write down the wavefunction, Ψ(x, t), t 0 in terms of the eigenstates. A linear operator acts on the system such that Aψ 1 = 2ψ 1 ψ 2 Aψ 2 = 2ψ 2 ψ 1 Aψ n = 0, n 3. Find the eigenvalues of A and obtain a complete set of normalised eigenfunctions, φ n, of A in terms of the ψ n. At time t = 0 a measurement is made and it is found that the observable corresponding to A has value 3. After time t, A is measured again. What is the probability that the value is found to be 1? 4/I/6B A particle moving in one space dimension with wavefunction Ψ(x, t) obeys the timedependent Schrödinger equation. Write down the probability density, ρ, and current density, j, in terms of the wavefunction and show that they obey the equation j x + ρ t = 0. The wavefunction is Ψ(x, t) = ( e ikx + R e ikx) e iet/, where E = 2 k 2 /2m and R is a constant, which may be complex. Evaluate j. Part IB 2007
33 /II/15B Let V 1 (x) and V 2 (x) be two real potential functions of one space dimension, and let a be a positive constant. Suppose also that V 1 (x) V 2 (x) 0 for all x and that V 1 (x) = V 2 (x) = 0 for all x such that x a. Consider an incoming beam of particles described by the plane wave exp(ikx), for some k > 0, scattering off one of the potentials V 1 (x) or V 2 (x). Let p i be the probability that a particle in the beam is reflected by the potential V i (x). Is it necessarily the case that p 1 p 2? Justify your answer carefully, either by giving a rigorous proof or by presenting a counterexample with explicit calculations of p 1 and p 2. 2/II/16B The spherically symmetric bound state wavefunctions ψ(r), where r = x, for an electron orbiting in the Coulomb potential V (r) = e 2 /(4πɛ 0 r) of a hydrogen atom nucleus, can be modelled as solutions to the equation d 2 ψ dr dψ r dr + a r ψ(r) b2 ψ(r) = 0 for r 0, where a = e 2 m/(2πɛ 0 2 ), b = 2mE/, and E is the energy of the corresponding state. Show that there are normalisable and continuous wavefunctions ψ(r) satisfying this equation with energies for all integers N 1. me 4 E = 32π 2 ɛ N 2 3/I/7B Define the quantum mechanical operators for the angular momentum ˆL and the total angular momentum ˆL 2 in terms of the operators ˆx and. Calculate the commutators [ˆL i, ˆL j ] and [ˆL 2, ˆL i ]. Part IB 2006
34 /II/16B The expression ψ A denotes the uncertainty of a quantum mechanical observable A in a state with normalised wavefunction ψ. Prove that the Heisenberg uncertainty principle ( ψ x)( ψ p) 2 holds for all normalised wavefunctions ψ(x) of one spatial dimension. [You may quote Schwarz s inequality without proof.] A Gaussian wavepacket evolves so that at time t its wavefunction is ψ(x, t) = (2π) 1 4 ( ) 1 ( i t exp x 2 4(1 + i t) Calculate the uncertainties ψ x and ψ p at each time t, and hence verify explicitly that the uncertainty principle holds at each time t. [ You may quote without proof the results that if Re(a) > 0 then ). and exp ( d dx exp ( x2 a ( x2 a ) ) x 2 exp ( x2 dx = 1 ( π ) 1 2 a 3 a 4 2 (Re(a)) 3 2 )) ( )) d ( ( dx exp x2 π ) 1 2 a dx = a 2 (Re(a)) 3 2 ]. 4/I/6B (a) Define the probability density ρ(x, t) and the probability current J(x, t) for a quantum mechanical wave function ψ(x, t), where the three dimensional vector x defines spatial coordinates. Given that the potential V (x) is real, show that J + ρ t = 0. (b) Write down the standard integral expressions for the expectation value A ψ and the uncertainty ψ A of a quantum mechanical observable A in a state with wavefunction ψ(x). Give an expression for ψ A in terms of A 2 ψ and A ψ, and justify your answer. Part IB 2006
35 /II/15G The wave function of a particle of mass m that moves in a onedimensional potential well satisfies the Schrödinger equation with a potential that is zero in the region a x a and infinite elsewhere, V (x) = 0 for x a, V (x) = for x > a. Determine the complete set of normalised energy eigenfunctions for the particle and show that the energy eigenvalues are E = 2 π 2 n 2 8ma 2, where n is a positive integer. At time t = 0 the wave function is ψ(x) = 1 5a cos ( πx ) + 2 sin 2a 5a ( πx ), a in the region a x a, and zero otherwise. Determine the possible results for a measurement of the energy of the system and the relative probabilities of obtaining these energies. In an experiment the system is measured to be in its lowest possible energy eigenstate. The width of the well is then doubled while the wave function is unaltered. Calculate the probability that a later measurement will find the particle to be in the lowest energy state of the new potential well. Part IB 2005
36 /II/16G A particle of mass m moving in a onedimensional harmonic oscillator potential satisfies the Schrödinger equation where the Hamiltonian is given by H Ψ(x, t) = i Ψ(x, t), t d 2 H = 2 2m dx m ω2 x 2. The operators a and a are defined by a = 1 ( βx + i ) 2 β p, a = 1 ( βx i ) 2 β p, where β = mω/ and p = i / x is the usual momentum operator. [a, a ] = 1. Show that Express x and p in terms of a and a and, hence or otherwise, show that H can be written in the form H = ( a a + 1 2) ω. Show, for an arbitrary wave function Ψ, that dx Ψ H Ψ 1 2 ω and hence that the energy of any state satisfies the bound E 1 2 ω. Hence, or otherwise, show that the ground state wave function satisfies aψ 0 = 0 and that its energy is given by E 0 = 1 2 ω. By considering H acting on a Ψ 0, (a ) 2 Ψ 0, and so on, show that states of the form (a ) n Ψ 0 (n > 0) are also eigenstates and that their energies are given by E n = ( n + 1 2) ω. Part IB 2005
37 /I/7G The wave function Ψ(x, t) is a solution of the timedependent Schrödinger equation for a particle of mass m in a potential V (x), H Ψ(x, t) = i Ψ(x, t), t where H is the Hamiltonian. Define the expectation value, O, of any operator O. At time t = 0, Ψ(x, t) can be written as a sum of the form Ψ(x, 0) = n a n u n (x), where u n is a complete set of normalized eigenfunctions of the Hamiltonian with energy eigenvalues E n and a n are complex coefficients that satisfy n a na n = 1. Find Ψ(x, t) for t > 0. What is the probability of finding the system in a state with energy E p at time t? Show that the expectation value of the energy is independent of time. 3/II/16G A particle of mass µ moves in two dimensions in an axisymmetric potential. Show that the timeindependent Schrödinger equation can be separated in polar coordinates. Show that the angular part of the wave function has the form e imφ, where φ is the angular coordinate and m is an integer. Suppose that the potential is zero for r < a, where r is the radial coordinate, and infinite otherwise. Show that the radial part of the wave function satisfies d 2 R dρ dr ρ dρ + (1 m2 ρ 2 ) R = 0, where ρ = r ( 2µE/ 2) 1/2. What conditions must R satisfy at r = 0 and R = a? Show that, when m = 0, the equation has the solution R(ρ) = k=0 A k ρ k, where A k = 0 if k is odd and if k is even. A k A k+2 = (k + 2) 2, Deduce the coefficients A 2 and A 4 in terms of A 0. By truncating the series expansion at order ρ 4, estimate the smallest value of ρ at which the R is zero. Hence give an estimate of the ground state energy. [You may use the fact that the Laplace operator is given in polar coordinates by the expression 2 = 2 r r r ] r 2 φ 2. Part IB 2005
38 /I/6G Define the commutator [A, B] of two operators, A and B. In three dimensions angular momentum is defined by a vector operator L with components L x = y p z z p y L y = z p x x p z L z = x p y y p x. Show that [L x, L y ] = i L z and use this, together with permutations, to show that [L 2, L w ] = 0, where w denotes any of the directions x, y, z. At a given time the wave function of a particle is given by ψ = (x + y + z) exp ( ) x 2 + y 2 + z 2. Show that this is an eigenstate of L 2 with eigenvalue equal to 2 2. Part IB 2005
39 /I/8D From the timedependent Schrödinger equation for ψ(x, t), derive the equation ρ t + j x = 0 for ρ(x, t) = ψ (x, t)ψ(x, t) and some suitable j(x, t). Show that ψ(x, t) = e i(kx ωt) is a solution of the timedependent Schrödinger equation with zero potential for suitable ω(k) and calculate ρ and j. What is the interpretation of this solution? 1/II/19D The angular momentum operators are L = (L 1, L 2, L 3 ). commutation relations and show that [L i, L 2 ] = 0. Let Write down their and show that L ± = L 1 ± il 2, L 2 = L L + + L L 3. Verify that Lf(r) = 0, where r 2 = x i x i, for any function f. Show that L 3 (x 1 + ix 2 ) n f(r) = n (x 1 + ix 2 ) n f(r), L + (x 1 + ix 2 ) n f(r) = 0, for any integer n. Show that (x 1 + ix 2 ) n f(r) is an eigenfunction of L 2 and determine its eigenvalue. Why must L (x 1 + ix 2 ) n f(r) be an eigenfunction of L 2? What is its eigenvalue? 2/I/8D A quantum mechanical system is described by vectors ψ = eigenvectors are ψ 0 = ( ) cos θ, ψ sin θ 1 = ( ) sin θ, cos θ with energies E 0, E 1 respectively. The system is in the state ( ) 0 is the probability of finding it in the state at a later time t? 1 ( ) a. The energy b ( ) 1 at time t = 0. What 0 Part IB 2004
40 /II/19D Consider a Hamiltonian of the form H = 1 ( )( ) p + if(x) p if(x), < x <, 2m where f(x) is a real function. Show that this can be written in the form H = p 2 /(2m) + V (x), for some real V (x) to be determined. Show that there is a wave function ψ 0 (x), satisfying a firstorder equation, such that Hψ 0 = 0. If f is a polynomial of degree n, show that n must be odd in order for ψ 0 to be normalisable. By considering dx ψ Hψ show that all energy eigenvalues other than that for ψ 0 must be positive. For f(x) = kx, use these results to find the lowest energy and corresponding wave function for the harmonic oscillator Hamiltonian H oscillator = p2 2m mω2 x 2. 3/I/9D Write down the expressions for the classical energy and angular momentum for an electron in a hydrogen atom. In the Bohr model the angular momentum L is quantised so that L = n, for integer n. Assuming circular orbits, show that the radius of the n th orbit is r n = n 2 a, and determine a. Show that the corresponding energy is then e2 E n =. 8πɛ 0 r n Part IB 2004
41 /II/20D A onedimensional system has the potential { 0 x < 0, V (x) = 2 U 2m 0 < x < L, 0 x > L. For energy E = 2 ɛ/(2m), ɛ < U, the wave function has the form a e ikx + c e ikx x < 0, ψ(x) = e cosh Kx + f sinh Kx 0 < x < L, d e ik(x L) + b e ik(x L) x > L. By considering the relation between incoming and outgoing waves explain why we should expect c 2 + d 2 = a 2 + b 2. Find four linear relations between a, b, c, d, e, f. Eliminate d, e, f and show that c = 1 D [ b + 1 ( λ 1 ) ] sinh KL a, 2 λ where D = cosh KL 1 2( λ + 1 λ) sinh KL and λ = K/(ik). By using the result for c, or otherwise, explain why the solution for d is d = 1 D [ a + 1 ( λ 1 ) ] sinh KL b. 2 λ For b = 0 define the transmission coefficient T and show that, for large L, T 16 ɛ(u ɛ) U 2 e 2 U ɛ L. Part IB 2004
42 /I/9A A particle of mass m is confined inside a onedimensional box of length a. Determine the possible energy eigenvalues. 1/II/18A What is the significance of the expectation value Q = ψ (x) Q ψ(x)dx of an observable Q in the normalized state ψ(x)? Let Q and P be two observables. By considering the norm of (Q + iλp )ψ for real values of λ, show that Q 2 P [Q, P ] 2. The uncertainty Q of Q in the state ψ(x) is defined as Deduce the generalized uncertainty relation, ( Q) 2 = (Q Q ) 2. Q P 1 2 [Q, P ]. A particle of mass m moves in one dimension under the influence of the potential 1 2 mω2 x 2. By considering the commutator [x, p], show that the expectation value of the Hamiltonian satisfies H 1 2 ω. 2/I/9A What is meant by the statement than an operator is hermitian? A particle of mass m moves in the real potential V (x) in one dimension. Show that the Hamiltonian of the system is hermitian. Show that d dt x = 1 m p, d dt p = V (x), where p is the momentum operator and A denotes the expectation value of the operator A. Part IB 2003
43 /II/18A A particle of mass m and energy E moving in one dimension is incident from the left on a potential barrier V (x) given by with V 0 > 0. V (x) = { V0 0 x a 0 otherwise In the limit V 0, a 0 with V 0 a = U held fixed, show that the transmission probability is T = (1 + mu 2 ) 1 2E 2. 3/II/20A The radial wavefunction for the hydrogen atom satisfies the equation 2 2m 1 r 2 d (r 2 ddr ) dr R(r) + 2 l(l + 1)R(r) e2 2mr2 Explain the origin of each term in this equation. R(r) = ER(r). 4πɛ 0 r The wavefunctions for the ground state and first radially excited state, both with l = 0, can be written as R 1 (r) = N 1 exp( αr) R 2 (r) = N 2 (r + b) exp( βr) respectively, where N 1 and N 2 are normalization constants. corresponding energy eigenvalues E 1 and E 2. Determine α, β, b and the A hydrogen atom is in the first radially excited state. It makes the transition to the ground state, emitting a photon. What is the frequency of the emitted photon? Part IB 2003
44 /I/9D Consider a quantum mechanical particle of mass m moving in one dimension, in a potential well, x < 0, V (x) = 0, 0 < x < a, V 0, x > a. Sketch the ground state energy eigenfunction χ(x) and show that its energy is E = 2 k 2 2m, where k satisfies k tan ka =. 2mV 0 k 2 2 [Hint: You may assume that χ(0) = 0. ] 1/II/18D A quantum mechanical particle of mass M moves in one dimension in the presence of a negative delta function potential 2 V = 2M δ(x), where is a parameter with dimensions of length. (a) Write down the timeindependent Schrödinger equation for energy eigenstates χ(x), with energy E. By integrating this equation across x = 0, show that the gradient of the wavefunction jumps across x = 0 according to lim ɛ 0 ( dχ dχ ) (ɛ) dx dx ( ɛ) [You may assume that χ is continuous across x = 0.] = 1 χ(0). (b) Show that there exists a negative energy solution and calculate its energy. (c) Consider a double delta function potential 2 V (x) = [δ(x + a) + δ(x a)]. 2M For sufficiently small, this potential yields a negative energy solution of odd parity, i.e. χ( x) = χ(x). Show that its energy is given by E = 2 2M λ2, where tanh λa = λ 1 λ. [You may again assume χ is continuous across x = ±a.] Part IB
45 /I/9D From the expressions L x = yp z zp y, L y = zp x xp z, L z = xp y yp x, show that (x + iy)z is an eigenfunction of L 2 and L z, and compute the corresponding eigenvalues. Part IB
46 /II/18D Consider a quantum mechanical particle moving in an upsidedown harmonic oscillator potential. Its wavefunction Ψ(x, t) evolves according to the timedependent Schrödinger equation, i Ψ t = 2 Ψ 2 2 x x2 Ψ. (1) (a) Verify that is a solution of equation (1), provided that Ψ(x, t) = A(t)e B(t)x2 (2) da dt = i AB, and db dt = i 2 2i B2. (3) (b) Verify that B = 1 2 tan(φ it) provides a solution to (3), where φ is an arbitrary real constant. (c) The expectation value of an operator O at time t is O (t) dxψ (x, t)oψ(x, t), where Ψ(x, t) is the normalised wave function. Show that for Ψ(x, t) given by (2), Hence show that as t, x 2 = 1 4Re(B), p2 = 4 2 B 2 x 2. x 2 p 2 4 sin 2φ e2t. [Hint: You may use dx e Cx2 x 2 dx e Cx2 = 1 2C.] Part IB
47 /II/20D A quantum mechanical system has two states χ 0 and χ 1, which are normalised energy eigenstates of a Hamiltonian H 3, with H 3 χ 0 = χ 0, H 3 χ 1 = +χ 1. A general timedependent state may be written Ψ(t) = a 0 (t)χ 0 + a 1 (t)χ 1, (1) where a 0 (t) and a 1 (t) are complex numbers obeying a 0 (t) 2 + a 1 (t) 2 = 1. (a) Write down the timedependent Schrödinger equation for Ψ(t), and show that if the Hamiltonian is H 3, then i da 0 dt = a 0, i da 1 dt = +a 1. For the general state given in equation (1) above, write down the probability to observe the system, at time t, in a state αχ 0 + βχ 1, properly normalised so that α 2 + β 2 = 1. (b) Now consider starting the system in the state χ 0 at time t = 0, and evolving it with a different Hamiltonian H 1, which acts on the states χ 0 and χ 1 as follows: H 1 χ 0 = χ 1, H 1 χ 1 = χ 0. By solving the timedependent Schrödinger equation for the Hamiltonian H 1, find a 0 (t) and a 1 (t) in this case. Hence determine the shortest time T > 0 such that Ψ(T ) is an eigenstate of H 3 with eigenvalue +1. (c) Now consider taking the state Ψ(T ) from part (b), and evolving it for further length of time T, with Hamiltonian H 2, which acts on the states χ 0 and χ 1 as follows: H 2 χ 0 = iχ 1, H 2 χ 1 = iχ 0. What is the final state of the system? Is this state observationally distinguishable from the original state χ 0? Part IB
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